Ramanujan–Nagell equation

In mathematics, in the field of number theory, the Ramanujan–Nagell equation is a particular exponential Diophantine equation.

Equation and solution

The equation is

\( 2^n-7=x^2 \, \)

and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15.

This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician WIlhelm Ljunggren, and subsequently proved shortly thereafter by the Norwegian mathematician Trygve Nagell. The values on n correspond to the values of x as:-

x = 1, 3, 5, 11 and 181[1]

Triangular Mersenne numbers

The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent: [2]

\( 2^b-1 = \frac{y(y+1)}{2} \)
\( \Leftrightarrow 8(2^b-1) = 4y(y+1) \)
\( \Leftrightarrow 2^{b+3}-8 = 4y^2+4y \)
\( \Leftrightarrow 2^{b+3}-7 = 4y^2+4y+1 \)
\( \Leftrightarrow 2^{b+3}-7 = (2y+1)^2 \)

The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan-Nagell numbers) are:

\( \frac{y(y+1)}{2} = \frac{(x-1)(x+1)}{8}\)

for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in OEIS).
See also

Scientific equations named after people

References

^ "Values of X corresponding to N in the Ramanujan-Nagell Equation". Wolfram MathWorld. Retrieved 2009-11-06.
^ Can N^2 + N + 2 Be A Power Of 2?, Math Forum discussion

S. Ramanujan (1913). "Question 464". J. Indian Math. Soc. 5: 130.
W. Ljunggren (1943). "Oppgave nr 2". Norsk Mat. Tidsskr. 25: 29.
T. Nagell (1948). "Løsning till oppgave nr 2". Norsk Mat. Tidsskr. 30: 62–64.
T. Nagell (1961). "The Diophantine equation x2+7=2n". Ark. Mat. 30: 185–187. doi:10.1007/BF02592006.
T.N. Shorey; R. Tijdeman (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. 87. Cambridge University Press. pp. 137–138. ISBN 0-521-26826-5.

Mathematics Encyclopedia